In this video, we will discuss the transition probability under harmonic potential. So let us consider the harmonic perturbation potential v of t that can be written in a very general form as shown here. This script v and v dagger guy here is the hermitian conjugate of v, and they contain all the dependence on the variables other than time. Position, momentum, spin, whatever that needs to be included is all contained here in script v. Only the time dependence is separated out and expressed by these e^i Omega t and e^negative i Omega t, these two terms shown here. Now, we can use the formula that we derived in the time-dependent perturbation theory for the 1st order coefficient c_n^1 and just plug in v of t expression here into that formula. We assume that the perturbation is turned on at t equals 0, then we simply obtain this equation. Now, this expression is similar to the constant perturbation case that we discussed in the previous video, except that Omega_ ni and Omega_ni is once again e_n minus e_i divided by h bar. That in the constant perturbation case is now replaced by Omega_ni plus or minus Omega here. Similarly to the constant perturbation case once again, as t goes to infinity, the probability, the absolute value square of this quotient is appreciable only when Omega_ni is equal to plus or minus Omega, or in terms of energy, the final energy, e_n is equal to the initial energy e_i plus or minus h bar Omega. The plus sign here corresponds to this term here, and the negative sign corresponds to this term. When the plus sign is satisfied, then this term becomes very large. When the minus sign is satisfied here, then this term becomes large. Now, let us consider the classical radiation field description. The Hamiltonian for the electron light interaction, electron radiation interaction is given as this. Here is the kinetic energy term for the electron, and this is the electrostatic potential, v is electrostatic potential, so this is the electrostatic potential energy, and this A here is the magnetic vector potential, and this term gives rise to the interaction with the magnetic field. In the non-relativistic case, we typically use the Coulomb gauge, which requires the divergence of A vector potential is 0 and we consider a monochromatic plane wave. Monochromatic means that we have a well-defined frequency Omega and the plane wave, meaning that the face front, the position of the constant phase is simply a plane. The plane wave is described by its simple, this sine or cosine function. We use a cosine function here. Here is the t vector, the wave vector of the plane wave and the magnitude of the wave vector is given by Omega over c, the dispersion relation in free space. Hat is the unit vector along the propagation direction. Omega, of course, is the angular frequency of the light. Epsilon hat is the polarization vector. It's the unit vector that specifies the direction of the field. A naught, of course, is the amplitude. You can see that this vector potential is divergence free. Divergence is 0 because the polarization, direction, and the propagation directions are chosen to be perpendicular to each other. Now we can plug in this expression for A into this, the last term of the Hamiltonian, A.p term, and we obtain this equation here. This equation you notice has the same mathematical form as the harmonic potential that we start out with in the previous slide. Here, with this harmonic potential, we can simply use the results of this perturbation theory in the previous slide by substituting the appropriate quantity, all these things with these exponential i Omega over c, n hat dot x up to this part, all of these becomes our script v. By substituting those quantities in force script to be, we can simply use the result of the time-dependent perturbation theory in the previous slide. That explicitly written here, so this is V sub ni dagger term is given by this. Then we can calculate the absorption rate by simply taking out the result of the time-dependent perturbation theory, calculate the probability function c sub n, absolute value squared, and take the time derivative, then we get the transition rate. Particularly we are calculating the absorption rate where the final energy is larger than the initial energy. If we choose the different sign for the energy conservation equation, then we will find the emission rate instead. Here, this is the transition rate that we can calculate using the time-dependent perturbation result. Once again, this last delta function here appears to simply impose energy conservation condition. But it needs to be understood that this result, the Fermi's Golden Rule as it is often called, is obtained by integrating over a range of energy for the final state. So if the final state is a continuum, as was the case in the example of [inaudible] recombination that we discussed in the previous lecture, then we simply integrate over energy using the density of state as we did in that case in the previous lecture. If the final state is a discrete energy level, even in that case, the linewidth cannot be infinitesimally small. There is some certain finite broadening because of the finite lifetime of that energy state, and we integrate over the line width, just like the continuum state, and that allows us to obtain this compact result. Now, it is convenient or useful to define absorption cross-section because it's something that's readily related to the experimental results. Absorption cross-section is defined as the absorbed energy, per time rate of energy absorption divided by the incident intensity, incident energy flux per unit area. If you divide those two, then the energy per time, energy per time cancels out, and this unit area remains and gives you a dimension of an area therefore, the cross-section. From the classical electrodynamic theory, energy flux is given as this. This C here is the speed of light, and this script U is the electromagnetic energy density. Electromagnetic energy density travels at the speed of light. That gives you the energy flux, flow of energy. That is given by this equation here. Now, then the absorption cross-section, by using the absorbed energy per time, which is defined by the transition rate that we just calculate in the previous slide, and that's in the denominator here. You multiply that by the light energy, which is H-bar Omega, that gives you the absorbed energy per time. Then you divide it by the incident flux per unit area, which is given by this energy flux equations here. You find this equation down here for the absorption cross-section. Now, let us expand the exponential function in the transition rate equation in the transition matrix element equation into Taylor series. We can do that as long as this exponent is small enough. Let's check whether the exponent actually is small enough. In a typical atomic transitions, the atomic energy level spacing is the energy of the radiation. You're making a transition induced by light between the atomic energy level. The light energy H-bar Omega should be related to roughly the atomic energy level spacing. The atomic energy level spacing is roughly given by this. This is basically the Z is the atomic number and E square is the charge of the electron in the nuclei and the radius of the atom. Now, let's calculate the wavelength, which is given by C over Omega wavelength of the light. From this, you can show that the wavelength of the light is roughly related to this. This C H-bar over E square is simply the inverse of defined structure constant 137. For R_atom divided by Lambda, which is this, Omega over C is 1 over Lambda, and X here, the position variable, would scale roughly as the radius of the atom. This exponent magnitude can be estimated to be this ratio, which is the atomic number divided by 137, is a small number. Indeed, it is justified to have this expansion and ignore higher-order term. The lowest order approximation will be just ignore everything and then just approximate this exponential function to be just one. Let's say that the exponential function is one, and then the transition matrix element here simply becomes this. This is the polarization unit vector, and this here is the momentum. Now the transition matrix element, it gets reduced to the momentum matrix element. Now, we can further show that this momentum matrix element is related to the position matrix element. For this, let's assume that the polarization is along the x-axis and the propagation direction is along the z-axis, then, while this E Epsilon hat is the x-unit vector. If you multiply that to your position, then it just pulls out the momentum. If you dot product it with the momentum with the x-unit vector, then it just pulls out the X component of the momentum. Using the uncertainty relationship, this X component of the momentum is related to the commutator X position. Do that but n and i are the eigenstates of H naught so that simply pulls out these energy eigenvalues for n and i state, which by dividing it by h bar gives you this Omega_ni. Now, you can show that this momentum matrix element is related to the position matrix element. Multiplying the electronic charge to this x displacement operator gives you electric dipole moment. This approximation is often called the electric dipole approximation, where the transition rate is determined by the matrix element of the electric dipole operator. In the electric dipole approximation, absorption cross-section is simply given by this. Alpha here is the fine structure constant, again. The trend shown rate is essentially determined by the absolute value square of the position matrix element or electric dipole matrix element, if you multiply the electronic charge to x. If the initial state is the ground state, then these Omega_ni is always positive because the final state energy is always greater than the initial state. Then we can simply integrate the absorption cross-section over all frequency and obtain this equation here. Now, let us define the oscillator strength f_ni. This is the strength of transition or strength of oscillator associated with the trend shown between these two states n and i. Let us define that as this, it is related to the transition rate, obviously, because this matrix element is related to the transition rate. Then we can simply see that summing this oscillator strength over all possible final states should be equal to 1. Why? Because, each of these terms, oscillator strength gives you the probability of making a transition to a certain state n. If you add these probabilities to all possible final states, the total probability should simply be 1. This is called the Thomas-Reiche-Kuhn sum rule, and the integration of the absorption cross-section then simply becomes this, and this is the oscillation sum rule that you can derive from the classical electrodynamics. This example here is one of the first examples where quantum mechanical results led to the correct classical result. Now before I finish, I just want to point out that we have discussed the absorption transitions only in this video. But there are two terms, if you recall, the perturbation potential has two terms. One term represents absorption process and the other term represents the emission process where the final energy is lower than the initial energy. You can apply the exact same thing for the other term to describe the emission process. This emission, light emission, induced by light illumination, you remember all of these transitions are induced by applying this time-harmonic dependent potential, which is produced by the light. This emission induced by light illumination is called the stimulated emission. There is, of course, a light emission not induced by instant light, which is called a spontaneous emission, and the spontaneous emission process is not described by the framework that we have discussed in this video.
